Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(B)$, where $B$ is the unit ball in $\mathbb{R}^n$. Prove or disprove the following.

Given any positive integer $k$, $$\sup_{|\alpha|=k,x\in B} |D^\alpha f(x)| \leq \sup_{r < 1} \lvert f^{(k)}(r) \rvert,$$ where $\alpha$ is a multi-index and $D^\alpha f$ is the corresponding derivative of $f$. By $f^{(k)}(r)$, we mean the $k^{th}$ derivative of $f$ as a function of $r=|x|.$

I try some functions, taking second order derivatives, and the inequality holds for all of them. The case where $k=1$ is easy to prove but I can't prove for a general $k$.

Instead of a general smooth $f$, can we prove the assertion for polynomials in which $f(r) = \sum_{j=0}^m c_j r^{2j}$ ?

PS: I asked this question on Math.SE but no one answered so it is posted here.